on an interval

SearchAdvancedHelp

(0.004 seconds)

21—30 of 142 matching pages

21: 24.19 Methods of Computation

… ►

A method related to “Stickelberger codes” is applied in Buhler et al. (2001); in particular, it allows for an efficient search for the irregular pairs $(2n,p)$. Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).

22: 8.18 Asymptotic Expansions of $I_x(a,b)$

… ►uniformly for $x\in (0,1)$ and $a/(a+b)$, $b/(a+b)\in [\delta ,1-\delta ]$, where $\delta$ again denotes an arbitrary small positive constant. …

23: 4.37 Inverse Hyperbolic Functions

… ►In (4.37.2) the integration path may not pass through either of the points $\pm 1$, and the function ${(t^2-1)}^{1/2}$ assumes its principal value when $t\in (1,\mathrm{\infty })$. … ►

4.37.19 $$\mathrm{arccosh}z=\ln \left(\pm {(z^2-1)}^{1/2}+z\right),$$ $z\in ℂ\setminus (-\mathrm{\infty },1)$,

ⓘ

Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $\setminus$: set subtraction and $z$: complex variable

Referenced by:

§4.37(iv), §4.37(iii), §4.37(iv)

Permalink:

http://dlmf.nist.gov/4.37.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

… ►It should be noted that the imaginary axis is not a cut; the function defined by (4.37.19) and (4.37.20) is analytic everywhere except on $(-\mathrm{\infty },1]$. … ►An equivalent definition is … ►

4.37.24 $$\mathrm{arctanh}z=\frac{1}{2}\ln \left(\frac{1+z}{1-z}\right),$$ $z\in ℂ\setminus (-\mathrm{\infty },-1]\cup [1,\mathrm{\infty })$;

ⓘ

Symbols:

$[a,b)$: half-closed interval, $ℂ$: complex plane, $\in$: element of, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, $\ln z$: principal branch of logarithm function, $(a,b]$: half-closed interval, $\setminus$: set subtraction, $\cup$: union and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

…

24: 18.35 Pollaczek Polynomials

… ►where, depending on $a,b,\lambda$, $D$ is a discrete subset of $ℝ$ and the $w_{\zeta }^{(\lambda )}(a,b)$ are certain weights. … ►

$w_{x_k}^{(\lambda )}(a,b)={\displaystyle \frac{{\rho }^{2k-1}{\left(1-{\rho }^2\right)}^{2\lambda +1}\mathrm{\Gamma }\left(2\lambda +k\right)}{2\mathrm{\Delta }k!}},$

… ►See Bo and Wong (1996) for an asymptotic expansion of $P_n^{(\frac{1}{2})}(\cos \left(n^{-\frac{1}{2}}\theta \right);a,b)$ as $n\to \mathrm{\infty }$, with $a$ and $b$ fixed. This expansion is in terms of the Airy function $\mathrm{Ai}\left(x\right)$ and its derivative (§9.2), and is uniform in any compact $\theta$-interval in $(0,\mathrm{\infty })$. Also included is an asymptotic approximation for the zeros of $P_n^{(\frac{1}{2})}(\cos \left(n^{-\frac{1}{2}}\theta \right);a,b)$. …

25: 13.20 Uniform Asymptotic Approximations for Large $\mu$

… ►uniformly with respect to $x\in (0,\mathrm{\infty })$ and $\kappa \in [0,(1-\delta )\mu ]$, where $\delta$ again denotes an arbitrary small positive constant. …

26: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

… ►uniformly with respect to $x\in (0,A]$ in each case, where $A$ is an arbitrary positive constant. … ►uniformly with respect to $\mu \in [0,(1-\delta )\kappa ]$ and $x\in (0,(1-\delta )(2\kappa +2\sqrt{{\kappa }^2-{\mu }^2})]$, where $\delta$ again denotes an arbitrary small positive constant. …

27: 1.10 Functions of a Complex Variable

… ►Assume that for each $t\in [a,b]$, $f(z,t)$ is an analytic function of $z$ in $D$, and also that $f(z,t)$ is a continuous function of both variables. …

28: 33.14 Definitions and Basic Properties

… ► $f(ϵ,\mathrm{\ell };r)$ is real and an analytic function of $r$ in the interval , and it is also an analytic function of $ϵ$ when . … ► $h(ϵ,\mathrm{\ell };r)$ is real and an analytic function of each of $r$ and $ϵ$ in the intervals and , except when $r=0$ or $ϵ=0$. …

29: 22.3 Graphics

… ►

►►

Figure 22.3.2: $k=0.7$, $-3K\le x\le 3K$, $K=1.8456\mathrm{\dots }$. For $\mathrm{cn}(x,k)$ the curve for $k=1/\sqrt{2}=0.70710\mathrm{\dots }$ is a boundary between the curves that have an inflection point in the interval $0\le x\le 2K\left(k\right)$, and its translates, and those that do not; see Walker (1996, p. 146). Magnify

…

30: 5.12 Beta Function

… ►where the contour starts from an arbitrary point $P$ in the interval $(0,1)$, circles $1$ and then $0$ in the positive sense, circles $1$ and then $0$ in the negative sense, and returns to $P$. …